University of Groningen

Strangeness photoproduction on the deuterium targetShende, Sugat Vyankatesh

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Chapter 2

Theory

The classification scheme of hadrons provided by the quark model will bediscussed in this chapter. Although Quantum Chromodynamics (QCD) is

known as the theory which explains the mechanism of the strong interaction,this theory is not easily applicable at medium excitation energy (1 GeV - 3 GeV)because at this energy the coupling constant becomes large. An introductionto reaction models like the partial-wave analysis and the isobar model, whichare used to extract the information from the experimental data, is given in thischapter. The calculations for the kaon photoproduction on the deuteron usingthe P-matrix approach and the isobar model are described.

2.1 Quark model

When a large number of subatomic particles was discovered, a classificationscheme for the particles was desired. In 1964 Murray Gell-Mann and Zweigintroduced the “quark”, a particle of which hadrons are composed. Quarks allowfor a new way of grouping particles in the quark model. A triplet of quarks i.e. up(u), down (d) and strange (s), was introduced forming the fundamental triplet(3) of the flavour SU(3) group. These quarks are spin 1

2fermions with isospin

component I3 = +12

for the up-quark, I3 = −12

for the down-quark and I3 = 0

for the strange-quark. The conjugate quark triplet (3) consists of antiquarks i.e.antiup (u), antidown (d) and antistrange (s) whose quantum numbers have theopposite sign as those of their quark counter parts. The hadrons are classifiedinto mesons and baryons. Therefore a new quantum number “Baryon number”was introduced. The baryons are assigned the baryon number = 1 and themesons are given a baryon number = 0. The strangeness content of hadrons isdefined by the quantum number “Strangeness”. In the quark model the relationbetween charge (Q), isospin (I3), baryon number (Bn) and strangeness (S) is

11

CHAPTER 2. THEORY 12

Flavour Spin Charge Isospin Bn S C B T

up (u) 1/2 2/3 1/2 1/3 0 0 0 0down (d) 1/2 -1/3 -1/2 1/3 0 0 0 0

strange (s) 1/2 -1/3 0 1/3 -1 0 0 0charm (c) 1/2 2/3 0 1/3 0 1 0 0bottom (b) 1/2 -1/3 0 1/3 0 0 -1 0

top (t) 1/2 2/3 0 1/3 0 0 0 1

Table 2.1: Summary of quark quantum numbers including baryon number (Bn),strangeness (S), charm (C), bottom (B), top (T ).

given by:

Q = I3 +Bn + S

2(2.1)

The quantity Bn +S = Y is called the hypercharge. Later subsequent discoveriesof heavier hadrons required a new class of quarks with flavour charm (c), bottom(b) and top (t). These additional quarks are associated with the correspondingquantum numbers, namely charm (C), bottom (B) and top (T ). The summaryof quantum numbers of quarks is given in table 2.1.

2.1.1 Mesons in the quark model

The mesons in the quark model are constructed from a quark-antiquark (qq) pairwhich can be described by the product of fundamental triplet and antitriplet,3⊗ 3. Combinations of qq from u, d and s quarks and antiquarks generate nonetsof mesons. The quantum numbers of these mesons are determined by the quan-tum numbers of the constituent quarks. The meson nonets are classified accord-ing to orbital angular momentum (L) and parity (P). Since quarks are spin 1

2

fermions, the spins of the qq state couple to give a total spin S = 0 or 1. Thetotal angular momentum of relative orbital angular momentum L and spin S ofthe qq states is J = L + S. The parity of the states is (−1)L+1. The nonet of thelightest mesons with JP = 0− is shown in figure 2.1. In this picture the mesonsare arranged according to the isospin component (I3) and strangeness (S) (formesons Y = S). The SU(3) nonet can be reduced to an SU(3) octet and a sin-glet state as in group theory the product 3⊗ 3 = 8⊕ 1 reduces to irreducible

13 2.1. QUARK MODEL

-

6

I3

S

−1 −12

0 12

1

−1

0

1

u uu u uu u

u u

K− K0

π− π0 η

η′π+

K0 K+

JP = 0−

Figure 2.1: The JP = 0− mesons arranged according to isospin (I3) and strangeness(S).

representations 8 and 1. The octet contains two isospin doublets with S = +1

and S = −1, one isospin triplet, and one isospin singlet with S = 0. The SU(3)singlet is an isospin singlet.

2.1.2 Baryons in the quark model

The baryons are constructed from the fundamental quark triplet by forming qqqstates. The lowest-lying baryon multiplets are formed with L = 0. The total spinof the three-quark state couples to S = 1

2and S = 3

2. This gives JP = 1

2

+ andJP = 3

2

+. In flavour SU(3) the multiplets are given by:

3⊗ 3⊗ 3 = 10S ⊕ 8Ms ⊕ 8Ma ⊕ 1A (2.2)

and three spin doublets in SU(2) can be combined as:

2⊗ 2⊗ 2 = 4S ⊕ 2Ms ⊕ 2Ma (2.3)

In these equations the subscript S denotes symmetric wavefunction, Ms and Ma

denotes mixed symmetric and mixed antisymmetric wavefunction and A denotesantisymmetric wavefunction with respect to interchange of the first two quarkflavours. For example, the direct product of three flavour triplets leads to a sym-metric decuplet, one octet of mixed symmetry and antisymmetry each, and asinglet of antisymmetric wavefunction. The direct product of three spin doublets

CHAPTER 2. THEORY 14

I3−1 −1

20 1

21

u uu uu u

u u

Ξ− Ξ0

Σ− Σ0

Λ0

Σ+

n p

JP = 12

+

-

6

I3

S

−32

−1 −12

0 12

1 32

−3

−2

−1

0 u u u uu u u

u uu

∆− ∆0 ∆+ ∆++

Σ∗− Σ∗0Σ∗+

Ξ∗− Ξ∗0

Ω−

JP = 32

+

Figure 2.2: The 12

+ octet and 32

+ decuplet states in the plane isospin (I3) vs strangeness(S).

gives the quadruplet of symmetric, a doublet of mixed symmetric and a doubletof mixed antisymmetric wavefunction. The combination of SU(3) flavour multi-plets and SU(3) spin multiplets provides the baryon multiplets. Figure 2.2 showsthe spectrum of baryons arranged according to isospin (I3) and strangeness (S).The baryon octet for JP = 1

2

+ and the decuplet for JP = 32

+ can be seen.A summary of the properties of meson and baryons involved in the study of

this thesis is given in table 2.2.

2.2 Quantum Chromodynamics

The quark model presented so far successfully classifies mesons and baryons.However, the ∆++ state in the baryon decuplet revealed a new problem. It iscomposed of three u quarks and since quarks are fermions this combination is notallowed by Pauli’s exclusion principle. This problem was solved by introducinganother degree of freedom called “colour charge”. The colour charge triplet isthe fundamental triplet of the colour SU(3) group. All the hadrons are coloursinglet states. The quark colour charge appears in three colours: red, green andblue (RGB). The antiquark carries anticolour: antired, antigreen and antiblue.The colour-anticolour combination of qq states forces mesons to colour singlets.

15 2.2. QUANTUM CHROMODYNAMICS

Particle Mass Quarks Life time Decay Branching(MeV) τ modes ratio (%)

K0S 497.648 ± 0.022 ds 0.8953 ±0.0006 π0π0 31.05 ± 0.14

×10−10 s π+π− 68.95 ± 0.14

Σ+ 1189.37 ± 0.07 uus 0.8018 ± 0.0026 pπ0 51.57 ± 0.30×10−10 s nπ+ 48.31 ± 0.30

p 938.27 uud > 2.1 × 1029 years − −

n 939.56 udd 885 ± 0.8 s pe−νe 100

Table 2.2: Fundamental properties of K0S , Σ+, proton (p) and neutron (n) [9].

In baryons the quarks carry a colour such that the baryons appear as coloursinglets.

The quarks are bound into the hadron because of the “strong interaction”.This strong interaction is mediated by “gluons”. According to colour SU(3) thegluons appear with eight colour-anticolour combinations. The gluons mediatethe interaction between the particles carrying colour charge. Since gluons carrycolour charge they not only couple to quarks but also to other gluons. This isan important difference to the electromagnetic interaction, where the photonshave no charge, and therefore can not couple with each other. Quantum Chro-modynamics (QCD) is the theory of the strong interaction which describes theinteraction between quarks and gluons.

The strong coupling constant αs depends on the momentum transfer Q2. Atvery high energies the coupling constant αs approaches zero, thus quarks canbe considered free. This is known as “asymptotic freedom”. The perturbationtheory can be applied to evaluate the QCD diagrams at very high energies. Atlower energies the coupling constant αs of the strong interaction approachesunity. The colour force between quarks becomes stronger. This effect is caused bythe self-coupling nature of the gluons. As the inter-quark distance increases, thepotential energy of the system increases in proportion to the distance and so thequarks and gluons can never be freed. This mechanism is called “confinement”.

CHAPTER 2. THEORY 16

In this case one can not rely on perturbation theory. Among the non-perturbativeapproaches to QCD, the most well established one is lattice QCD [27]. Thisapproach evaluates the QCD interaction numerically on a discrete set of space-time points (called the lattice). This approach results in extensive numericalcomputations which require supercomputers. Lattice QCD has wide applicabilityand holds great promise for the future.

2.3 Model calculations

Apart from the three quark constituents of the baryon, the QCD Lagrangian givesrise to a large number of quark-antiquark pairs inside the nucleon, called the“sea” quarks. Because the gluon can couple to gluons, there is also a large num-ber of gluons turning into gluon-gluon combinations within the sea. The simplequark models treat the nucleon as if it consists of three non-sea quarks only. Toaccount for the quark sea, the three effective quarks are assumed to have a spa-tial extent and much larger mass. These new approximated quarks are calledconstituent quarks.

Although the quark models and QCD calculations predict the baryon reso-nances, an experiment can not directly verify these states. An experiment onlyobserves a particular final state, and one cannot easily differentiate the eventsproduced via resonances from non-resonant mechanisms. An experiment canonly provide the information on the cross sections, polarizations and other ob-servables. The basic task is to extract information on the resonances with theirproperties like mass, width, spin and parities. To obtain this information fromthe experimental data one uses theoretical models like the isobar model or modelindependent analysis methods like partial-wave analysis.

Partial-wave analysis

The very strong point of partial-wave analysis is that it makes use of the oper-ator decomposition method which provides a tool for a universal constructionof partial wave amplitudes for reactions with two- and many-body final states.The incident particles are described by a plane wave. After the collision, in ad-dition to the plane wave there is an outgoing spherical wave originating at thescattering center. The scattering amplitude is represented by the partial-wave ex-pansion. The cross section is given by the Breit-Wigner resonance formula (seesection 9.2.3 in ref. [28]) which is a function of the mass and the width of the

17 2.4. THE HYPERON-NUCLEON INTERACTION

resonance. The identification of the nucleon resonance is carried out by usinginput parameters from available photo- and pion- induced reaction data in differ-ent final states. If the final states are stemming from the same resonance, thenthe resonance must have the same mass, total width and coupling constants,in all reactions under study. This imposes strong constraints for the analysis.Different weights are assigned to different data sets depending on the statisti-cal accuracy. For example, high statistics data should carry a high weight. Themaximum-likelihood fit is performed on the data under study using the numberof resonances, their spin and parity and the relative weights of the different datasets. Recently a partial-wave analysis of data on photoproduction of πN , ηN ,KΛ and KΣ final states was published [29]. They presented the evidence forfour new resonances which have not been found before.

Isobar models

Isobar models assume that the particle production and decay proceed via reso-nances and all subsequent decays appear to be two-body reactions. For example,the decay A → B + C + D is actually A → X + D followed by X → B + C.The intermediate state X is a resonance state with mass and width. Each two-body process has an associated Lorentz-invariant matrix element that representsthe angular distribution. The associated amplitude of each two-body processis treated as a complex constant. A complex constant is a reasonable approxi-mation for narrow resonances. Generally, isobar models assume that only res-onances below a certain cut-off angular momentum contribute to low-energynuclear processes. The resonances with a spin above 2 are usually ignored. Inpractice, only resonances with less than a cut-off spin J are used and radial ex-citations are frequently ignored [30]. A more detailed discussion of the isobarmodel actually used in theoretical predictions follows in section 2.5.

2.4 The hyperon-nucleon interaction

In order to understand the behavior of the strong interaction inside hadrons it isessential to explore the multiplet structure of hadrons. It is unavoidable to knowthoroughly not only the nucleon-nucleon (NN) interaction but also the hyperon-nucleon (YN) interaction for Λ and Σ hyperons. Experimental information onthe YN interaction is rather scarce. Knowledge about the YN interaction can be

CHAPTER 2. THEORY 18

gained from: (1) the study of hyperfragments, (2) scattering experiments and(3) the final-state interaction.

Hypernuclei, i.e. atomic nuclei in which one or more baryons containing astrange quark (hyperons) have been incorporated. The formation of hypernu-clear states in most experiments involves a strangeness exchange reaction likeK− +X → Y + π− with X = p, n and Y = Λ,Σ [36] and π+ + AZ → K+ + A

ΛZ∗

[37]. These processes are carried through complex pion interactions and presentdifficulties to extract details about the YN potential.

For YN scattering, experimental data are available from scattering (Λp →Λp,Σ±p→ Σ±p), charge exchange (Σ−p→ Σ0n) and inelastic processes (Σ−p→Λn,Λp → Σ0p) [38, 39, 40]. The results from these experiments suffer fromstatistical accuracy and disagree with potential parameters as derived from thestudy of hypernuclei [19].

The YN interaction in the framework of final-state interaction (FSI) may pro-vide unique information, because this is the only way to produce directly an YNinteraction. The reaction involved in this case is

(γ or π) + d→ K + Y +N (2.4)

The study of these reactions had a lot of attention after the pioneering study ofRenard and Renard [17]. To reveal the YN interaction, a number of approacheshad been attempted including the Nijmegen soft-core one-boson exchange (OBE)potential [41], Yukawa-type potential [19], plane-wave approximation[21]. Allthese models mostly concentrate on K+ production which involves either a Λ ora Σ± hyperon. The study presented in this thesis involves K0 photoproductionon the deuteron and the associated Σ+ hyperon production. The selection ofthis channel has the advantage that Λ− Σ0 conversion needs not be considered.Recently some effort has been invested to calculate the γd → K0Σ+n reaction[23]. In the subsequent section the predictions by [21, 23] on the YN interactionwill be discussed.

2.5 Isobar model

The kaon photoproduction on the deuteron is given by the reaction:

γ(pγ) + d(pd) → K(pK) + Y (pY ) +N(pN) (2.5)

19 2.5. ISOBAR MODEL

where pγ, pd, pK , pY and pN denote the four-momenta of the photon, deuteron,kaon, hyperon and nucleon, respectively. The general expression for the kaonproduction cross section is given in [43] as:

dσ =δ4(pγ + pd − pK − pY − pN)mNmY d

3pNd3pY d

3pK

48(2π)5|~vγ − ~vd|EγEdENEYEK

×∑

µY µNµdλ

|MKγdµY µNµdλ(~pY , ~pN , ~pK , ~pd, ~pγ)|2 (2.6)

where µY , µN , µd and λ denote the spin projections of hyperon, nucleon, deuteronand the photon polarization, respectively. MKγd

µY µNµdλ is the kaon photoproduc-tion amplitude.

The differential cross section of kaon photoproduction on the deuteron iscalculated by integrating the kaon momentum between pmin

K and pmaxK (pK is

bound for a given photon energy) as:

dσ

dΩK

=

∫ pmaxK

pminK

dpK

∫dΩ∗

Y N κ∑

µY µNµdλ

|MKγdµY µNµdλ(~pY N , ~pK , ~pγ)|2 (2.7)

with the kinematic factor

κ =mYmNp

2Kp

∗Y N

24(2π)2pγEKWY N

(2.8)

In this expression, Ω∗Y N and p∗Y N are the solid angle and relative momentum

in the Y N center-of-mass system, respectively. The term WY N =√E2

Y N − ~p2Y N

gives the invariant-mass of the Y N system.

2.5.1 The photoproduction amplitude

The kaon photoproduction amplitude on the nucleon was approximated by thetree diagrams shown in figure 2.3. The tree diagram is a connected Feynman dia-gram which represents the possible reaction mechanism (figure 2.3). Three pos-sible tree diagrams based on Mandelstam variables are called s−, t−, u− channeldiagrams. The s-channel is equivalent to the centre-of-mass energy of the reac-tion. For example the two particles ( 1 and 2, i.e. γ and N in figure 2.3) formthe intermediate particle that eventually decays into final particles (3 and 4, i.e.K and Y in figure 2.3). The u-channel and t-channel correspond to the square ofthe four-momentum transfer. In the u-channel, the particle 1 (γ) emits an inter-

CHAPTER 2. THEORY 20

Figure 2.3: Feynman diagrams for the electromagnetic production of kaons on the nu-cleon. Electromagnetic vertices are denoted by (a), (b) and (c), hadronic vertices by (1),(2) and (3) (taken from [31]).

mediate particle (Y, Y ∗) and becomes particle 4 (Y ). The intermediate particlethen is absorbed by particle 2 (N) and becomes particle 3 (K). The u-channeldiagram is given by the second graph in figure 2.3. In the t-channel diagram, theparticle 1 (γ) becomes particle 3 (K) by exchanging an intermediate particle,which is then absorbed by particle 2 (N) which becomes particle 4 (Y ). Thet-channel diagram is shown as the third graph in figure 2.3.

All observables are determined by the photoproduction amplitudeMKγdµY µNµdλ,

which is the matrix element of a corresponding photoproduction operator MKγd,as:

MKγdµY µNµdλ(~pY N , ~pK , ~pγ) = 〈~pY N~pKµY µN |MKγd|~pγµdλ〉 (2.9)

In these calculations the kaon photoproduction operator is composed of two-body subsystems of the final state i.e. impulse approximation, YN rescattering,KN rescattering and a two-step pion mediated process. Figure 2.4 shows thedifferent diagrams representing these four processes contributing to the kaonphotoproduction operator. The kaon photoproduction operator is given as:

MKγd = MKγdIA + MKγd

Y N + MKγdKN + MKγd

Kπ (2.10)

where MKγdIA ,MKγd

Y N ,MKγdKN and MKγd

Kπ denote the operators for impulse approx-imation, hyperon-nucleon rescattering, kaon-nucleon rescattering and the pionmediated process, respectively.

21 2.5. ISOBAR MODEL

Figure 2.4: The diagrams representing kaon photoproduction on the deuteron, usedin the calculations by A. Salam. Diagram (a): impulse approximation (IA), (b): YNrescattering, (c): KN rescattering and (d): πN → KY process.

The impulse approximation

In the impulse approximation the incoming photon interacts with one nucleoninside the deuteron producing a kaon and hyperon, while the other nucleonremains untouched, i.e., it acts as a spectator. Then the transition amplitudeMKγd

IA in equation 2.10 is given by:

MKγdIA,µY µNµdλ = 〈~pY N~pKµY µN |MKγN |~pγµdλ〉 (2.11)

where the µ’s denote the spin projections of the corresponding particles and λ thephoton polarization. The symbol MKγN is the elementary kaon photoproductionoperator which is given as:

MKγN = uµY

(4∑

i=1

AKγNi Γi

λ

)uµN

(2.12)

where uµYand uµN

are the hyperon and nucleon Dirac spinors, respectively. Theterm Γi

λ is the invariant Dirac operator and AKγNi is the invariant amplitude. The

equation 2.11 has been solved by A. Salam in [43].

CHAPTER 2. THEORY 22

YN-rescattering

In YN rescattering, the photon interacts with one nucleon inside the deuteron byphotoproducing the kaon and the hyperon. Then the hyperon interacts withthe other nucleon stimulating YN rescattering effects. For the calculation ofYN rescattering, the diagrams (a) and (b) in figure 2.4 are considered i.e. theimpulse approximation summed with the YN rescattering contribution. Then thephotoproduction operator in this case is given by:

MKγdIA+Y N = MKγd

IA + MKγdY N (2.13)

= MKγdIA + TY N GY NMKγd

IA (2.14)

where TY N is the YN-scattering operator and GY N is the free hyperon-nucleonoff-shell propagator. The term TY N obeys the Lippmann-Schwinger equation andcontains the hyperon-nucleon potential. This potential is given by the Nijmegeninteraction potential described in terms of one-boson exchanges. After solvingequation 2.14, one can obtain the YN rescattering amplitude by subtraction ofthe impulse approximation.

KN-rescattering

In KN rescattering, the photoproduced kaon interacts with another nucleon in-side the nucleus as shown in diagram (c) of figure 2.4. Unlike YN rescattering,KN rescattering contributions are calculated directly. The corresponding ampli-tude is given by:

MKγdKN,µY µNµdλ = 〈~pY N~pKµY µN |TKN GKNMKγN |~pγµdλ〉 (2.15)

where TKN is the kaon-nucleon scattering operator and GKN is the free kaon-nucleon propagator. This equation can be directly solved to get the contributionsfrom KN rescattering.

γd→ πNN → KYN process

In this process, first the photon interacts with one of the nucleons inside thedeuteron producing a pion and a nucleon i.e. pion photoproduction. Then thephotoproduced pion interacts with another nucleon to produce a kaon and ahyperon (diagram (d) in figure 2.4). For this πN → KY process the amplitude is

23 2.5. ISOBAR MODEL

constructed similar to that of KN-rescattering. The amplitude has been obtainedby replacing the kaon photoproduction operator in equation 2.15 by the pionphotoproduction operator and the KN scattering amplitude by the πN → KY

transition amplitude, i.e.:

MKγN → MπγN (2.16)

TKN → TπK (2.17)

where TπK and MπγN denote the transition operators for the pion-mediated kaonprocess and the pion photoproduction on the deuteron, respectively.

2.5.2 Results from the Isobar-model calculations

The results are calculated by using the deuteron wave function of the Bonnmodel [47]. For these calculations the KAON-MAID framework [48] was used.The coupling constants were determined by fitting the elementary model to theSAPHIR data [49, 50].

The results of the calculations are shown in figure 2.5 and 2.6. The total crosssection as a function of the incident photon energy is shown in figure 2.5. Thecross section for the impulse approximation is shown by the dotted line. Accord-ing to these calculations the effects due to additional YN (short-dashed line) andKN (dashed line) rescattering are negligible, therefore the corresponding curvesare lying on top of each other. The curve for the πN → KN process (solid line)shows a strong enhancement in the cross section.

Figure 2.6 (left) shows the differential cross section as a function of the hy-peron angle θ′Y for incident photon energy, Eγ = 1.3 GeV, forward kaon angle,θK = 1, and high kaon momentum, pK = 810 MeV/c. The angle θ′Y is the hy-peron angle measured relative to the direction of momentum transfer ~pγ − ~pK inthe deuteron rest frame (see figure 2.7). For this observable these calculationspredict a significant effect due to YN rescattering towards higher hyperon angleθ′Y . Figure 2.6 (right) shows the differential cross section as a function of kaonangle θK for incident photon energy Eγ = 1.14 GeV.

During the work presented in this thesis, the SAPHIR collaboration has pub-lished new results on proton channels, i.e. the γp → K+Λ, γp → K+Σ0 andγp → K0Σ+ [12, 13]. These new results are more precise and cover all angulardistributions in the center-of-mass energy range up to W = 2.4 GeV. The cou-pling constants in the Isorbar Model were refitted by using these new SAPHIR

CHAPTER 2. THEORY 24

Figure 2.5: The total cross section as function of the incoming photon energy calculatedby A. Salam (taken from [23]). The calculations are shown for (a) impulse approxima-tion (dotted line), (b) YN rescattering (short-dashed line), (c) KN rescattering (dashedline) and (d) πN → KN process (solid line). Since the curves for impulse approxima-tion, YN and KN rescattering are lying on top of each other, they are indistinguishable.

25 2.6. P-MATRIX APPROACH

Figure 2.6: Left: Differential cross section as function of hyperon angle θ′Y calculatedby A. Salam (taken from [23]) for Eγ = 1.3 GeV, θK = 1 and pK = 810 MeV/c. Thecalculations are shown for (a) impulse approximation (dotted line), (b) YN rescattering(short-dashed line), (c) KN rescattering (dashed line) and (d) πN → KY process (solidline). The curves for YN rescattering, KN rescattering and two-step process are lying ontop of each other, therefore they are indistinguishable. Right: Differential cross sectionas function of kaon angle θK (taken from [23]) for Eγ = 1.14 GeV.

results. The newly fitted cross sections are shown in ref. [44]. These refittedvalues nicely describe not only the measurements for the γp → K0Σ+ channelbut also for the γp→ K+Λ channel, and for the channel γp→ K+Σ0 the modelreproduces the measurements up to 2 GeV. Using these recently fitted values,the new predictions on the K0 photoproduction on the deuteron are available inref.[42] which are used for comparing the results obtained from the experimentdiscussed in this thesis.

2.6 P-matrix approach

Kerbikov [21] has calculated the inclusive differential cross section for the γd→K+Y n, (Y = Λ,Σ0) reaction in the framework of a P-matrix approach. The P-matrix approach was described by Jaffe and Lee [32] and establishes the connec-tion between the scattering data and the multi-quark states. This formalism hasbeen successfully applied in meson-meson, meson-baryon and baryon-baryonscattering [33]. In these calculations the inclusive double-differential cross sec-

CHAPTER 2. THEORY 26

Figure 2.7: Kinematical representation of the hyperon angle θ′Y . The angle θ′Y is thehyperon angle measured relative to the direction of momentum transfer ~pγ−~pK , definingthe z’ axis.

27 2.6. P-MATRIX APPROACH

tion is given by:

d2σ

d|pK |dΩK

=1

211π5

p2K

kMdEK

λ1/2(s2,m2Y ,m

2n)

s2

∫dΩ∗

Y n|T |2 (2.18)

where, k is the four-momentum of the photon. Md is the mass of the deuteron.The parameters pK , EK ,ΩK represent kaon momentum, energy and solid anglein the deuteron rest (laboratory) frame. The term Ω∗

Y n is the solid angle ofthe Y n-system in the center-of-mass system. The quantity λ is the standardkinematical function λ(x, y, z) = x2 − 2(y + z)x − (y − z)2. The s2 is given bys2 = (pY + pn)2. In this equation T is the kaon photoproduction amplitude forthe γd→ K+Y n reaction.

The amplitude T was approximated by the contribution of two types of di-agrams. First, the tree diagrams (plane-wave approximation) (figure 2.3) andsecond, the triangle diagram with YN rescattering. More details on these calcu-lations can be found in [21].

In plane-wave approximation an elementary amplitude was derived from thetree level effective Lagrangian taking into account several resonances in the s, tand u channel [22]. The amplitude for the tree diagram is given in [21] as:

T = [2(2π)3Md]1/2MγKψd (2.19)

where, MγK is the elementary photoproduction amplitude on a proton. ψd is therelativistic deuteron wave function.

For the triangle diagrams the amplitude T is derived as:

T = 2µ[(2π)32Md]1/2 ×

∫dpn

(2π)3

MγKψd(pn)F (p, p′;E ′)

p2 − p′2(2.20)

where, p and p′ are the Y n center-of-mass momenta before and after the FSI,F (p, p′;E ′) is the Y n amplitude derived from coupling between hadron andquark channels via a non-local energy-dependent potential.

The results from this calculation are shown in figure 2.8. This figure showsthe double-differential cross section (equation 2.18) as a function of incidentphoton energy at |pK | = 1.091 GeV/c and θK = 0. In this picture the dashedcurve is the plane-wave approximation without FSI. The ΛN production thresh-old is at Eγ = 1.4 GeV. The plane-wave approximation including FSI is shown bythe solid curve. The YN interaction shows a strong growth of the cross sectionclose to the ΛN threshold. One can also see a strong enhancement in the cross

CHAPTER 2. THEORY 28

Figure 2.8: Double-differential cross section as function of incident photon energy(taken from [21]) for pK = 1.091 GeV/c and θK = 0. In this picture the result ofthe plane-wave approximation PWA (dashed curve) and PWA including YN FSI (solidcurve) is shown.

section close to the ΣN threshold at Eγ = 1.49 GeV. This contradicts the conclu-sion drawn from a previous analysis [19] that FSI were only important close toΛN threshold and insignificant at higher energies.

The kaon production experiments presented in this thesis, therefore, shouldprovide a solid testing ground to establish the importance of final-state interac-tions.